A160078 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in binary expansion reversed). Binary analog of Lychrel numbers.

22, 26, 28, 35, 37, 41, 45, 46, 47, 49, 60, 61, 67, 75, 77, 78, 84, 86, 89, 90, 93, 94, 95, 97, 105, 106, 108, 110, 116, 120, 122, 124, 125, 131, 135, 139, 141, 147, 149, 152

Offset: 1

Author

Dremov Dmitry (dremovd(AT)gmail.com), May 01 2009

Keywords

• base
• nonn

Comments

Number of iterations equals 1000, but all non-seeded numbers (under) fall out in 32 iterations

Examples

			22 = 10110
10110 + 01101 = 100011
100011 + 110001 = 1010100...
Not forming a palindrome after 1000 iterations.
		

Links

• Diofant.ru, Problem: binary Lychrel numbers under 1024. (Russian language!) [From Dremov Dmitry (dremovd(AT)gmail.com), May 03 2009]

Crossrefs

Binary version of A023108.

Programs

Python

from sympy.ntheory.digits import digits
def make_int(l, b):
    return int(''.join(str(d) for d in l), b)
maxn = 102
it = []
for i in range( 1, maxn ) :
    d = digits( i, 2 )[1:]
    isLychrel = True
    for j in range( 1000 ) :
        d = digits( make_int( d, 2 ) + make_int( d[::-1], 2 ), 2 )[1:]
        if d == d[::-1] :
            it.append( j + 1 )
            isLychrel = False
            break
    if isLychrel :
        it.append( 0 )
print('Maximum iterations for non-seed numbers', max( it ))
Lychrel = []
for i in range( len(it) ) :
    if it[i] == 0 :
        Lychrel.append( i + 1 )
print('Count of binary Lychrel numbers', len( Lychrel ))
print('All binary lichler under', maxn)
print('Decimal form', Lychrel)
print('Binary form', list(map( lambda x: ''.join( map( str, toSystem( x, 2 ) ) ), Lychrel )))